First slide

Operations of matrix

Question

If $E (θ) = [\begin{array}{cc} \cos^{2} θ & \cos θsin θ \\ \cos θsin θ & \sin^{2} θ \end{array}]$ and $θ$ and $ϕ$ differ by an odd multiple of $\frac{π}{2}$ , then E ( $θ$ ) E ( $ϕ$ ) is a

Easy

A

null matrix
B

unit matrix
C

diagonal matrix
D

none of these

Solution

$E (θ) = [\begin{array}{cc} \cos^{2} θ & \cos θsin θ \\ \cos θsin θ & \sin^{2} θ \end{array}] θ - ϕ = \frac{π}{2} \Rightarrow θ = \frac{π}{2} + ϕ E (ϕ) = [\begin{array}{cc} \sin^{2} ϕ & - \cos ϕ \sin ϕ \\ - \cos ϕ \sin ϕ & \cos^{2} ϕ \end{array}] E (θ) E (ϕ) = [\begin{array}{cc} \cos^{2} θ & cosθsinθ \\ cosθsinθ & \sin^{2} θ \end{array}] [\begin{array}{cc} \sin^{2} ϕ & - \cos ϕ \sin ϕ \\ - \cos ϕ \sin ϕ & \cos^{2} ϕ \end{array}] = \begin{matrix} [\begin{matrix} \cos^{2} θ s {in}^{2} ϕ - \cos^{2} {θsin}^{2} ϕ & - \cos^{2} θ s {in}^{2} ϕ + \cos^{2} {θsin}^{2} ϕ \\ \cos^{2} θ s {in}^{2} ϕ - \cos^{2} {θsin}^{2} ϕ & - \cos^{2} θ s {in}^{2} ϕ + \cos^{2} {θsin}^{2} ϕ \end{matrix}] \end{matrix} = [\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}]$

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